Name the two players who have won the Champions League, Premier League, FA Cup, League Cup and European Championships?

Peter Schmeichel

yep, Schmeichel is one

International Record 1991-2001
129 Caps for Denmark - 1 Goal
1992 European Championship Winner
Honours with United
1999 European Champions League
1999 F.A. Premier League
1999 F.A. Cup
1997 F.A. Premier League
1996 F.A. Cup
1996 F.A. Premier League
1994 F.A. Cup
1994 F.A. Premier League
1993 F.A. Premier League
1992 League Cup

manutdzone.com/legends/PeterSchmeichel.htm no longer exists
https://www.transfermarkt.com/peter-schmeichel/erfolge/spieler/3465

The other one might be Anelka, if he was in the squad when Real Madrid won the Champions League.

It's not Anelka, he's missing a League Cup medal.

Nicolas Anelka Career
PSG (France), Arsenal (England), Real Madrid (spain), PSG (France), Liverpool (England), Manchester City (England), Fenerbahce (Turkey)
Nicolas Anelka Honours
2001 Confederations Cup winner (France)
2000 European Championship winner (France)
2000 Champions League winner (Real Madrid)
1998 Premiership Champions (Arsenal)
1998 FA Cup winner (Arsenal)

123football.com/players/a/nicolas-anelka/index.htm nolonger exists
https://www.transfermarkt.com/nicolas-anelka/profil/spieler/3226

Wasn't Anelka on loan to Liverpool when they won the League Cup?

Liverpool won the League Cup in 2001 and 2003, but Anelka wasn't there (the Finals were played in Feb/March).

Anelka was on loan to Liverpool during the second half of the 2001-02 season, when he played 12 games for PSG, and finished the season at Liverpool, playing 20 games. His debut was on Boxing Day 01 away to Villa, and his last game was on 11th May 02 at home to Ipswich Town.